Linear Systems over Finite Abelian Groups

We consider a system of linear constraints over any finite Abelian group G of the following form: i(x1, . . . , xn) ≡ i,1x1 + · · · + i,nxn ∈ Ai for i = 1, . . . , t and each Ai ⊂ G, i,j is an element of G and xi’s are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MODq boolean function, when the order of G and q are co-prime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS’09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential bounds on the size of boolean depth-four circuits of the form MAJ◦AND◦ANYO(1)◦ MODm for computing the MODq function, when m, q are co-prime. No superpolynomial lower bounds were known for such circuits for computing any explicit function. This ...

• Type: Journal
• Year: 2011
• Where: COCO
• Authors: Arkadev Chattopadhyay, Shachar Lovett
• Keywords: Abelian Group, Algorithms, COCO 2011, Distinct Primes, Linear Constraints